#!/usr/local/bin/perl

# ----------------------------------------------------------------
# John Kerl
# kerl.john.r@gmail.com
# 2005-03-29
#
# This is an example of numerical path integration.  Here, the path integral in
# x and y is analytically reduced to a 1-D integral in t, then numerically
# estimated.
#   C : x^2 + y^2 = 9, counterclockwise closed loop.
#   x  =  3 cos t   y  = 3 sin t
#   dx = -3 sin t   dy = 3 cos t
#   path integral -2y dx + x^2 dy
#   f(x,y) = -2y   g(x,y) = x^2
#   path integral f(x,y) dx + g(x,y) dy
#   = integral f(x,y) dx/dt dt + g(x,y) dy/dt dt
#   = integral from 0 to 2 pi of 18 sin^2 t + 27 cos^3 t dt
# Analytically, one expects 18 pi.
# ----------------------------------------------------------------

$pi  =  4.0 * atan2(1.0,1.0);
$tlo =  0.0;
$thi =  2.0 * $pi;

$nt  = 4000;
$nt  = $ARGV[0] if (@ARGV);
$dt  = ($thi - $tlo) / $nt;

# Alternatively:
#$dt  = 0.001;

$sum = 0.0;
for ($t = $tlo; $t < $thi; $t += $dt) {
	$ht = 18 * sin($t)**2 + 27 * cos($t)**3;
	$sum += $ht * $dt;
}
printf "%11.6f\n", $sum;